Solve the following inequation and represent the solution set on the number line: 3x/5 + 2 < x + 4 ≤ x/2 + 5, x ∈ R

Given:

3x/5 + 2 < x + 4 ≤ x/2 + 5, x ∈ R

First inequality:

3x/5 + 2 < x + 4

3x/5 − x < 4 − 2

−2x/5 < 2

Multiplying by 5:

−2x < 10

Dividing by −2 reverses the inequality:

x > −5 ...(1)

Second inequality:

x + 4 ≤ x/2 + 5

x − x/2 ≤ 5 − 4

x/2 ≤ 1

x ≤ 2 ...(2)

Combining (1) and (2):

−5 < x ≤ 2

Solution set: {x ∈ R : −5 < x ≤ 2}

On the number line: open circle at −5, closed circle at 2, and the interval between them is shaded.

Solve the two inequalities separately and take their common interval. The endpoint −5 is excluded because the inequality is strict; 2 is included because the inequality is ≤.